Kinematics in one dimension

1. Point particle

A body is described as a point particle when its dimensions are negligible compared with the distances traveled, so that its position is specified by a single point. A car that travels $100\mathrm{km}$ of highway is a point particle; the same car maneuvering in a parking lot a few meters wide is not. Kinematics describes the motion of the point particle through position, velocity and acceleration, without dealing with the causes of motion, which are the subject of dynamics (we refer to the topic on Newton’s laws).

2. Reference frame

A motion is rectilinear if the point moves along a straight line. To describe it, one fixes a reference frame consisting of

• an $x$-axis superimposed on the line of motion, with an origin $O$, a positive orientation and a unit of length, the meter;
• a clock that measures the time $t$ in seconds starting from a chosen initial instant.

The position of the point at the instant $t$ is its coordinate $x$ on the axis, positive or negative depending on the side of $O$ on which it lies. The origin, the orientation and the initial instant are chosen freely, and a suitable choice simplifies the calculations: for a falling stone the axis can be oriented upward with the origin at the ground, or downward with the origin at the starting point. Position, displacement, velocity and acceleration are vectors, and in rectilinear motion they all have the direction of the axis; therefore each of them is described by its component along the axis alone, a signed number. For vectors in the plane we refer to the topic on motion in the plane.

3. Equation of motion

The equation of motion is the function $t\mapsto x(t)$ that assigns to each instant the position of the point. Its graph in the $(t,x)$ plane, with time on the horizontal axis and position on the vertical axis, is the position–time graph. The position–time graph is different from the trajectory, which in rectilinear motion is a segment of the $x$-axis; the graph additionally shows when the point occupies each position. For example the law $x(t)=4t-t^2$, with $x$ in meters and $t$ in seconds, describes a point that starts from the origin, advances to $x=4\mathrm{m}$ at the instant $t=2\mathrm{s}$ and then turns back, passing through the origin again at the instant $t=4\mathrm{s}$. In a law written with numbers only, the coefficients have the units that make the terms homogeneous; here $4m/s$ and $1m/{\mathrm{s}}^2$.

4. Displacement and distance traveled

The displacement of the point between the instants $t_1$ and $t_2$ is the difference of the positions

It is positive if the final position lies on the positive side of the axis relative to the initial one, negative in the opposite case, and it depends only on the two extreme positions. The distance traveled is the total length of the path, and it coincides with $|\Delta x|$ only if the point never reverses the orientation of its motion between $t_1$ and $t_2$. For the law $x(t)=4t-t^2$ between $0$ and $5\mathrm{s}$ the displacement is $x(5)-x(0)=-5\mathrm{m}$, whereas the distance traveled is $4\mathrm{m}$ on the way out plus $9\mathrm{m}$ on the way back, that is, $13\mathrm{m}$.

1. Average velocity

The average velocity between the instants $t_1$ and $t_2>t_1$ is the ratio of the displacement to the time interval:

It is measured in $m/s$; to go from $km/h$ to $m/s$ one divides by $3.6$, since $1km/h=\frac{1000\mathrm{m}}{3600\mathrm{s}}$. The sign of $v_m$ indicates the orientation of the overall displacement, and in the position–time graph $v_m$ is the slope of the secant line through the points $(t_1,x(t_1))$ and $(t_2,x(t_2))$. The ratio of the distance traveled to the time taken is called the average speed and is always positive. For the law $x(t)=4t-t^2$ between $0$ and $5\mathrm{s}$ one has $v_m=\frac{-5\mathrm{m}}{5\mathrm{s}}=-1m/s$, whereas the average speed is $\frac{13\mathrm{m}}{5\mathrm{s}}=2.6m/s$.

2. Instantaneous velocity

The instantaneous velocity at the instant $t$ is the limit of the average velocity over shorter and shorter time intervals starting at $t$:

that is, the derivative of the equation of motion. In the position–time graph $v(t)$ is the slope of the tangent line. Therefore the point moves in the positive orientation of the axis where $x(t)$ increases ($v>0$), in the negative orientation where $x(t)$ decreases ($v<0$), and is momentarily at rest where the tangent is horizontal ($v=0$). The absolute value $|v|$ measures how fast the point moves, and it is the value shown by the speedometer. For $x(t)=4t-t^2$ one has $v(t)=4-2t$, so $v(0)=4m/s$, $v(2)=0$ and $v(5)=-6m/s$; at the instant $t=2\mathrm{s}$ the point reverses the orientation of its motion.